the Levi-Civita symbol

The Levi-Civita symbol is a bookkeeping device: a set of numbers, each either +1, -1, or 0, labelled by two or more indices, that encodes the single idea of 'antisymmetry' -- swap any two labels and the sign flips. In three dimensions it is written epsilon_ijk with i, j, k each running over 1, 2, 3. It is the machine that turns the messy component formulas for cross products, curls, and determinants into one compact line.

Precisely, epsilon_ijk equals +1 when (i,j,k) is an even permutation of (1,2,3) -- that is (1,2,3), (2,3,1), (3,1,2); equals -1 for an odd permutation -- (3,2,1), (1,3,2), (2,1,3); and equals 0 whenever any two indices are equal. With the Einstein summation convention (a repeated index is summed), the cross product is (a x b)_i = epsilon_ijk a_j b_k and the curl is (curl F)_i = epsilon_ijk partial_j F_k. The key identity epsilon_ijk epsilon_ilm = delta_jl delta_km - delta_jm delta_kl (with delta the Kronecker delta) is what proves the 'BAC-CAB' rule a x (b x c) = b(a . c) - c(a . b) in a single line.

Strictly, in flat Cartesian coordinates epsilon_ijk is a 'symbol' -- a fixed array of numbers, the same in every such frame. It is NOT a true tensor under general coordinate changes: the genuine tensor is the Levi-Civita tensor, which equals sqrt(g) times the symbol, where g is the determinant of the metric. That extra sqrt(g) is exactly what makes volume elements and cross products transform correctly in curvilinear or curved coordinates. Confusing the symbol with the tensor is a common and costly slip the moment you leave Cartesian axes.